Title: Intersection theory on logarithmic Picard group
Abstract: Let C/B be a family of nodal curves of genus g with n sections.
The relative Jacobian parametrizing multi-degree 0 line bundles is not
proper over the base B. Classically, one can compactify the relative
Jacobian using stable rank 1 torsion-free sheaves, which depends on a
choice of stability condition. Recently, Molcho and Wise constructed the
logarithmic Picard group, providing a canonical compactification. In this
talk, we introduce the logarithmic tautological subring of the logarithmic
Picard group inside the logarithmic Chow group. We prove that the proper
pushforward to the base preserves the logarithmic tautological ring.
When the base is the moduli space of stable curves, we propose a
conjectural closed formula for the pushforward of monomials of divisor
classes. This conjecture is motivated by an explicit computation of
Arinkin’s kernel over the compactified Jacobians. Over the open substack of
compact type or integral curves, we prove the conjecture using the Fourier
transformation. This is a joint work in progress with Samouil Molcho and
with Aaron Pixton.
Title: Intersection theory on logarithmic Picard group
Abstract: Let C/B be a family of nodal curves of genus g with n sections.
The relative Jacobian parametrizing multi-degree 0 line bundles is not
proper over the base B. Classically, one can compactify the relative
Jacobian using stable rank 1 torsion-free sheaves, which depends on a
choice of stability condition. Recently, Molcho and Wise constructed the
logarithmic Picard group, providing a canonical compactification. In this
talk, we introduce the logarithmic tautological subring of the logarithmic
Picard group inside the logarithmic Chow group. We prove that the proper
pushforward to the base preserves the logarithmic tautological ring.
When the base is the moduli space of stable curves, we propose a
conjectural closed formula for the pushforward of monomials of divisor
classes. This conjecture is motivated by an explicit computation of
Arinkin’s kernel over the compactified Jacobians. Over the open substack of
compact type or integral curves, we prove the conjecture using the Fourier
transformation. This is a joint work in progress with Samouil Molcho and
with Aaron Pixton.